CHAPTER 3
Proper proximality and first`2-Betti numbers
mild technical assumption.
Theorem 3.1.1. LetΓbe a countable exact group. Ifβ1(2)(Γ)>0, thenΓis properly proximal.
One concrete class of groups that satisfy the assumption of Theorem 3.1.1 is the class of one relator groups with at least 3 generators [Gue02, DL07], which was not known to be properly proximal before.
Since weak amenability implies exactness (see e.g. [Kir95, Proposition 2] and [Oza07]), Theorem 3.1.1 together with [BIP21, Theorem 1.5] implies that for a weakly amenable groupΓwithβ1(2)(Γ)>0,LΓhas no Cartan subalgebras andL∞(X)oΓhas a unique Cartan subalgebra, up unitary conjugacy, for any action free ergodic p.m.p.Γy(X,µ), which recovers the result in [PV14a] concerning groups with positive first`2-Betti numbers. Although it should be noted that [BIP21, Theorem 1.5] follows the same general strategy as laid out in [PV14a].
Our approach to Theorem 3.1.1 is rather indirect. In fact, we first obtain the following cocycle superrigid- ity result for non-properly proximal groups, from which Theorem 3.1.1 follows in combination with [PS12, Corollary 1.2].
Theorem 3.1.2. LetΓbe a countable group,(X0,µ0)be a diffuse standard probability space andΓy(X0Γ,µ0Γ) =:
(X,µ)be the Bernoulli action. IfΓis exact and contains a nonamenable non-properly proximal wq-normal subgroup, thenΓy(X,µ)is{T}-cocycle superrigid, i.e., any1-cocycle w:Γ×X→Tis cohomologus to a homomorphism.
Another theme that we explore is the rigidity of Bernoulli shifts of non-properly proximal group. Much of the work is heavily inspired by Popa’s pioneering work on Bernoulli shifts of rigid groups [Pop06c, Pop06d, Pop07a].
Theorem 3.1.3. LetΓ be a countable group with infinite conjugacy classes (i.c.c.), (X0,µ0)be a diffuse standard probability space andΓy(X0Γ,µ0Γ) =:(X,µ)be the Bernoulli action. LetΛbe a countable group andΛy(Y,ν)a free ergodic p.m.p. action such that L∞(Y)oΛ∼= (L∞(X)oΓ)t for some0<t≤1. IfΛ is exact and contains a nonamenable, non-properly proximal normal subgroup, then t=1 andΓyX and ΛyY are conjugate.
In particular, it follows that the fundamental group ofL∞(X)oΓis trivial ifΓis a countable, nonamenable, i.c.c., exact and non-properly proximal group andΓyXis the Bernoulli action. Furthermore, since proper proximality and exactness are stable under measure equivalence [IPR19, Oza07], Theorem 3.1.3 also implies the following OE-superrigidity result.
Theorem 3.1.4. LetΓbe a countable nonamenable i.c.c. group(X0,µ0)be a diffuse standard probability space andΓy(X0Γ,µ0Γ) =:(X,µ)be the Bernoulli action. IfΓis exact and non-properly proximal, then
ΓyX is OE-superrigid, i.e., if a free ergodic p.m.p. actionΛy(Y,ν)is orbit equivalent toΓy(X,µ), then these two actions are conjugate.
As a consequence, every countable nonamenable i.c.c. exact group has at least one desirable rigidity property, depending on whether or not it is properly proximal: either every group measure space II1factor has at most one weakly compact Cartan subalgebra, or else Bernoulli shifts are OE-superrigid.
All the above theorems are derived from the following von Neumann algebraic statement.
Theorem 3.1.5. LetΓbe a countable group,(X0,µ0)be a diffuse standard probability space andΓy(X0Γ,µ0Γ) =:
(X,µ)be the Bernoulli action. SupposeΓis exact and N⊂M:=L∞(X)oΓis a von Neumann subalgebra that has no amenable and no properly proximal direct summand. Then there exists a s-malleable deformation {αt}t∈Ron M such thatαt→idN uniformly on the unit ball of N, as t→0.
Here, the s-malleable deformation is in the sense of Popa [Pop06a, Pop06c] and this specific deformation is the one associated with Gaussian actions [Fur07] (see Section 3.4 for details). Proper proximality is for von Neumann algebras, in the sense of [DKEP22]. We note that Theorem 3.1.2 follows from Theorem 3.1.5 together with Popa’s seminal work on cocycle superrigidity [Pop07a, Pop08], and Theorem 3.1.3 is a com- bination of Theorem 3.1.5 and with Popa’s conjugacy criterion for Bernoulli actions [Pop06d]. Exactness of groups is crucial to our proof as we exploit the fact thatZoΓis biexact relative toΓ, providedΓis exact.
Let us finish with some comparisons between our results and some existing results on inner-amenable groups. The family of exact, non-properly proximal groups is strictly larger than the family of exact, inner- amenable groups, due to [IPR19], [DTDW20] and [GHW05], as well as the permanence properties of exact- ness of groups (see e.g. [BO08, Section 5.1]). Moreover, exactness and proper proximality are both stable under measure equivalence and W∗-equivalence [Oza07, IPR19], while inner-amenability is not preserved under measure equivalence [DTDW20] and is not known to be stable under W∗-equivalence. Therefore, The- orem 3.1.2 can be seen as a generalization of [TD20, Theorem 11] in the case ofT-valued cocycles associated with Bernoulli shifts of exact groups. And under the mild assumption on exactness, Theorem 3.1.1 general- izes [Dri22, Corollary F] and Theorem 3.1.5 extends [Dri22, Theorem E] in the case of wreath products.
Comments on the proofs.Let us outline the proof of Theorem 3.1.5, which uses the recently developed notion of proper proximality in [BIP21, IPR19, DKEP22] and Popa’s deformation/rigidity theory. The proof is divided into three steps. First we observe in Proposition 3.2.3 that for any von Neumann subalgebraNin L(ZoΓ), withΓexact, ifNhas no amenable direct summand, then it must be properly proximal relative toLΓ in the sense of [DKEP22]. This is a direct adaptation of [DKEP22, Theorem 7.1], sinceZoΓis biexact relative toΓ[BO08, Proposition 15.3.6]. Next in Section 3.3, we use techniques from [DKE22], which extends the idea in [DKE21, Lemma 3.3] to the von Neumann algebra setting. Continuing in the above setting with
N properly proximal relative toLΓ, we show thatN either has a properly proximal direct summand or is amenable relative toLΓinsideL(ZoΓ). In this step, the notion of normal bidual developed in [DKEP22, Section 2] is extensively used. Lastly, using a technique from [Ioa15], we conclude in Section 3.4 that if N⊂L(ZoΓ)is amenable relative toLΓ, thenN must be rigid with respect to the s-malleable deformation {αt}associated withL(ZoΓ). Altogether, we obtain that ifN⊂L(ZoΓ)has no amenable or properly proximal direct summand, thenNmust beαt-rigid.